According to OPIC projections, a driver who has had a policy for between two and three years should be able to save about 5.2 percent when they switch carriers. That percentage grows as time passes, with customers who have stayed with a company for between eight and nine years able to save an average of 19 percent. However, obtaining those savings in full may require policyholders with multiple lines of coverage to switch all of their coverage to the new insurer.

"We encourage all policyholders to frequently test the market by seeking quotes," OPIC executive director Deeia Beck said in a statement. "There is no downside."

The OPIC report acknowledged that shopping can be "difficult or tedious" for consumers. A July J.D. Power and Associates report shed some light on that shopping experience, finding that the rate that consumers looked around for auto coverage dropped to its lowest point in five years.

But that didn't mean attractive deals weren't out there—the same study found that shoppers switched their insurer at a historic rate and that the average shopper switching their carrier saved more than $300.

Policyholders may be in longer-term relationships with their insurer because of high satisfaction levels, according to a consumer survey from Deloitte, which found that nearly 9 out of every 10 respondents reported satisfaction with service while 8 out of every 10 reported satisfaction with price.

That same survey found that only about 1 out of every 5 surveyed said they compared quotes in the past year while only about 1 out of every 4 said they checked out the competition at least once every two years.